Possible Models

We denote the obesity outcome for subject \(i\) by \(y_i\), the gender for subject \(i\) by \(x_{1i}\), the race for subject \(i\) by \(x_{2i}\), and the age for subject \(i\) by \(x_{3i}\).

A logistic regression model for (prevalence of) obesity as a function of gender, race and age including all possible interaction terms is \[ \begin{eqnarray*} y_i & \sim & \mbox{Bernoulli}\left\{p_i\right\} \\ \mbox{logit}\left\{p_i\right\} & = & \alpha_0 + f_{g}(x_{1i},{\bf \beta}) + f_{r}(x_{2i},{\bf \gamma}) + f_{a}(x_{3i},{\bf \delta}) + \\ & & f_{g,r}(x_{1i},x_{2i},{\bf \beta\gamma}) + f_{g,a}(x_{1i},x_{3i},{\bf \beta\delta}) + f_{r,a}(x_{2i},x_{3i},{\bf \gamma\delta}) + \\ & & f_{g,r,a}(x_{1i},x_{2i},x_{3i},{\bf \beta\gamma\delta}) \\ \end{eqnarray*} \]

For modeling, gender must be coded with \(1~(=2-1)\) numeric variable, and race must be coded with \(4~(=5-1)\) numeric variables, and age will be represented by 5 variables, the basis for the natural cubic spline with 5 df or 6 knots (more generally, \(k\) variables, the basis for the natural cubic spline with \(k\) df or \(k+1\) knots).

The two-way interaction between gender and race is encoded using the \(4 = 1 \times 4\) products of the \(1\) variable used to encode gender and the \(4\) variables used to encode race. The two-way interaction between gender and age is encoded using the \(5 = 1 \times 5\) products of the \(1\) variable used to encode gender and the \(5\) variables used to encode age. The two-way interaction between race and age is encoded using the \(20 = 4 \times 5\) products of the \(4\) variables used to encode race and the \(5\) variables used to encode age.

The three-way interaction between gender, race and age is encoded using the \(20 = 1 \times 4 \times 5\) products of the \(1\) variable used to encode gender, the \(4\) variables used to encode race and the \(5\) variables used to encode age.

The model with all of these terms uses \[59 = 2 \times 5 \times 6 - 1 = 1 + 4 + 5 + 4 + 5 + 20 + 20\] df (regression parameters excluding the intercept).

There are seven simplifications of this model that could be of substantive interest: 1. an additive model for gender, race and age with \(10 = 1 + 4 + 5\) df \[ \begin{eqnarray*} \mbox{logit}\left\{p_i\right\} & = & \alpha_0 + f_{g}(x_{1i},{\bf \beta}) + f_{r}(x_{2i},{\bf \gamma}) + f_{a}(x_{3i},{\bf \delta}) \\ \end{eqnarray*} \] * the effect of gender does not vary by race and/or age * the effect of race does not vary by gender and/or age * the effect of age does not vary by gender and/or race 2. a two-way interaction between gender and race plus an additive effect of age with \(14 = 1 + 4 + 5 + 4\) df \[ \begin{eqnarray*} \mbox{logit}\left\{p_i\right\} & = & \alpha_0 + f_{g}(x_{1i},{\bf \beta}) + f_{r}(x_{2i},{\bf \gamma}) + f_{g,r}(x_{1i},x_{2i},{\bf \beta\gamma}) + f_{a}(x_{3i},{\bf \delta}) \\ \end{eqnarray*} \] * the effect of gender varies by race, but not by age * the effect of race varies by gender, but not by age * the effect of age does not vary by gender and/or race 3. a two-way interaction between gender and age plus an additive effect of race with \(15 = 1 + 4 + 5 + 5\) df \[ \begin{eqnarray*} \mbox{logit}\left\{p_i\right\} & = & \alpha_0 + f_{g}(x_{1i},{\bf \beta}) + f_{r}(x_{2i},{\bf \gamma}) + f_{a}(x_{3i},{\bf \delta}) + f_{g,a}(x_{1i},x_{3i},{\bf \beta\delta}) \\ \end{eqnarray*} \] * the effect of gender varies by age, but not by race * the effect of race does not vary by gender and/or age * the effect of age varies by gender, but not by race 4. a two-way interaction between age and race plus an additive effect of gender with \(30 = 1 + 4 + 5 + 20\) df \[ \begin{eqnarray*} \mbox{logit}\left\{p_i\right\} & = & \alpha_0 + f_{g}(x_{1i},{\bf \beta}) + f_{r}(x_{2i},{\bf \gamma}) + f_{a}(x_{3i},{\bf \delta}) + f_{r,a}(x_{2i},x_{3i},{\bf \gamma\delta}) \\ \end{eqnarray*} \] * the effect of gender does not vary by race and/or age * the effect of race varies by age, but not by gender * the effect of age varies by race, but not by gender 5. a two-way interaction between gender and race plus a two-way interaction between gender and age with \(19 = 1 + 4 + 5 + 4 + 5\) df \[ \begin{eqnarray*} \mbox{logit}\left\{p_i\right\} & = & \alpha_0 + f_{g}(x_{1i},{\bf \beta}) + f_{r}(x_{2i},{\bf \gamma}) + f_{a}(x_{3i},{\bf \delta}) + \\ & & f_{g,r}(x_{1i},x_{2i},{\bf \beta\gamma}) + f_{g,a}(x_{1i},x_{3i},{\bf \beta\delta}) \\ \end{eqnarray*} \] * the effect of gender varies by race and age * the effect of race varies by gender, but not by age * the effect of age varies by gender, but not by race 6. a two-way interaction between race and gender plus a two-way interaction between race and age with \(34 = 1 + 4 + 5 + 4 + 20\) df \[ \begin{eqnarray*} \mbox{logit}\left\{p_i\right\} & = & \alpha_0 + f_{g}(x_{1i},{\bf \beta}) + f_{r}(x_{2i},{\bf \gamma}) + f_{a}(x_{3i},{\bf \delta}) + \\ & & f_{g,r}(x_{1i},x_{2i},{\bf \beta\gamma}) + f_{r,a}(x_{2i},x_{3i},{\bf \gamma\delta}) \\ \end{eqnarray*} \] * the effect of gender varies by race, but not by age * the effect of race varies by gender and age * the effect of age varies by race, but not by gender 7. a two-way interaction between gender and age plus a two-way interaction between race and age with \(35 = 1 + 4 + 5 + 5 + 20\) df \[ \begin{eqnarray*} \mbox{logit}\left\{p_i\right\} & = & \alpha_0 + f_{g}(x_{1i},{\bf \beta}) + f_{r}(x_{2i},{\bf \gamma}) + f_{a}(x_{3i},{\bf \delta}) + \\ & & f_{g,a}(x_{1i},x_{3i},{\bf \beta\delta}) + f_{r,a}(x_{2i},x_{3i},{\bf \gamma\delta}) \\ \end{eqnarray*} \] * the effect of gender varies by age, but not by race * the effect of race varies by age, but not by gender * the effect of age varies by gender and race

The usefulness of these potential simplifications largely depends on the goals of our study. For example, if racial comparisons (the effects of race) are of primary interest, model 3 (the effect of race does not vary by gender and/or age), model 5 (the effect of race varies by gender, but not by age), and model 7 (the effect of race varies by age, but not by gender) represent meaningful simplifications of the full model with a three-way interaction, while the other models are not.

Assessing Adequacy of Modeling Assumptions

Before using the model, we must evaluate the adequacy of the modeling assumptions. For logistic regression models, the only assumptions are correct structural model and independence (which cannot be evaluated based on the observed data). The only concern is the potential for inadequacies in the representation of the effect of age, e.g. insufficient df. We can evaluate this assumption using plots of the residuals against age by gender and/or race.

PROC SGPANEL DATA=Tmp;
  PANELBY Gender / LAYOUT=ROWLATTICE;
    REFLINE 0 / AXIS=Y;
*  PBSPLINE X=Age Y=Residual / JITTER; *large samples;
  LOESS X=Age Y=Residual / JITTER; *small smaples;
RUN;

PROC SGPANEL DATA=Tmp;
  PANELBY Race / LAYOUT=ROWLATTICE;
    REFLINE 0 / AXIS=Y;
*  PBSPLINE X=Age Y=Residual / JITTER; *large samples;
  LOESS X=Age Y=Residual / JITTER; *small smaples;
RUN;

PROC SGPANEL DATA=Tmp;
  PANELBY Race Gender;
    REFLINE 0 / AXIS=Y;
*  PBSPLINE X=Age Y=Residual / JITTER; *large samples;
  LOESS X=Age Y=Residual / JITTER; *small smaples;
RUN;